The third mass protests against the government's higher education plans took place today as thousands of students took to the streets despite the freezing weather.

Large demonstrations took place in Brighton, Birmingham, Bristol, Manchester, Newcastle, Oxford and London. The Metropolitan police said 153 arrests had been made in the capital, 146 of which occurred after a group refused to leave Trafalgar Square at the end of the demonstration in London. Windows were smashed and missiles thrown at police, who charged at protesters with batons.

Students climbed on rooftops, stormed council buildings and stopped traffic in dozens of town centres, many saying they hoped the display of feeling would reverberate in Westminster.

Earlier there had been chaotic scenes in the capital when 4,000 students marching toward parliament tried to evade what they believed were attempts by police to "kettle" them in the bleak weather.

The demonstrators responded by dispersing across the city in separate marches, leading police in cat-and-mouse chases. One "feeder" march headed into the City, while others meandered past bemused onlookers at Oxford Circus and Hyde Park Corner, and near Buckingham Palace, stopping traffic en route.

"This is one of the most bizarre demonstrations I have been on," said Michael Chessum, 21, as he jogged up Regent Street with a group of riot police in tow. "It has been a shambolic policing operation because we agreed with them beforehand that we would march along Whitehall – but the spirit and determination of the students to get their point across has been pretty impressive."

The Met denied it had intended to kettle protesters, despite evidence of metal barriers and rows of officers waiting along Whitehall. It blamed the confusion on protesters, who, the force said, had begun their march earlier than agreed. "We made sure we had a flexible plan and sufficient resources to enable people to come back to Trafalgar Square where the protest was due to be held," said Chief Inspector Jane Connors. "That is what we did, moving around London, encouraging people to come back and meet together. We wanted to minimise disruption."

The mood was more harmonious elsewhere, although in Brighton about 600 protesters marched through the city before trying to force their way into Hove town hall. About 100 people managed to scale the roof of a car park and threw missiles, said police, but there were no arrests. Students also scaled a roof in Liverpool, where there were two arrests.

In Newcastle, students occupying a university building marched through the city centre in a peaceful event. Northumbria police said in a statement they had "nothing but praise" for the campaigners. "There were no arrests and no reports of any trouble of any kind," the force said.

Ten people were arrested in Bristol when about 1,000 protesters from both universities lit flares and pelted police with mustard. The M32 was closed when it seemed that the march might go towards the motorway.

In Birmingham, about 40 protesters broke into a council hall building, prompting a standoff with security and police. There were similar scenes in Leeds, where about 40 students occupied a university building, and in Oxford, where students invaded the county council offices.

Video footage showed protesters entering the Oxford building and walking through corridors before being ejected by police. The Conservative leader of the local authority, Keith Mitchell, said on Twitter: "County Hall invaded by an ugly, badly dressed student rabble. God help us if this is our future."

Greater Manchester police said there were five arrests in the city, but that only a "loose cordon" of officers was placed around hundreds of demonstrators who had gathered in the city centre.

About 400 students also walked peacefully through Cambridge, and, in Edinburgh, 300 protesters marched along the Royal Mile in the city and gathered at a rally outside the Scottish parliament. There was also an impromptu sit-in at Queens University, Belfast, and at the Trent building on Nottingham University's campus.

The scale and reach of this month's student protests have shocked the authorities, who fear that mobilisation against cuts could spread. Riot police were called to Lewisham town hall on Monday night when 100 protesters in the London borough tried to force their way into a meeting where councillors were voting to cut the budget by £60m. Police said arrests were made and several officers received minor injuries. The same protest groups are expected to focus on a council meeting in Camden, north London, tomorrow.

Many of the protests were organised by students occupying up to 32 university buildings across Britain. They have largely been independent of the National Union of Students. Threatened with a no-confidence vote, the NUS president, Aaron Porter, recently apologised for the union's "spineless" caution toward student activism and promised more support.